The Carleman’s Inequality for a Negative Power Number

By the method of indeterminate coefficients we prove the inequality ∞ n=1 ar 1 + ar 2 +· · ·+ar n n 1r ≤   1 − r −1 r ∞ n=1 an if −1 ≤ r < 1 r = 0 r r − 1 2 r−1 r ∞ n=1 an if r < −1, where an ≥ 0, n = 1 2 ∞ n=1 an < ∞.